Some Results On Counting I_π-Characters

In this thesis, the problem for counting irreducible π-partial characters (the so called I_π-characters) in a π-separable group was studied, and the following four main results were obtained:Theorem 1 Let (G, N, θ) and (H, M, φ) be I_π-character triples such that NH = G and N ∩ H = M. If θ_M =φ, then |I_π(G|θ)| = |I_π(H|φ)|.The result solved the counting problem of I_π-character triples under the condition of character restriction.Theorem 2 Let G be a 7r-separable group and suppose that G = NU, where N is a normal subgroup of G and U is a subgroup with π'-index in G. If θ ∈ I_π(N) satisfies the condition that θ_N∩U is irreducible, then for each x∈ I_π(G|θ), xu is irreducible. Furthermore, the restriction x → Xu defines a bijection from I_π(G|θ) onto I_π(U|φ).The above theorem generalized and strengthened the related results due to Isaacs and Navarro.Theorem 3 Let (G,N,θ) be an I_π-character triple. Then |I_π(G|θ)| = |G : N|_π if and only if the following three conditions hold:(a) a Hall π-subgroup of G/N is abelian,(b) a Hall π'-subgroup of G/N is normal, and(c) θ is extendible to H, where H/N is a Hall π-subgroup of G/N.The result solved the maximal value problem for counting I_π-characters triples.Theorem 4 Let (G, N, θ) be an I_π-character triple and let H/N be a Hall π-subgroup of G/N. If every Hall π'-subgroup of G/N is normal, then xh is irreducible for all x ∈ I_π(G|θ ). Furthermore, the restriction x→ Xh defines a bijection from I_π(G|θ) onto I_π(H|θ). In particular, |I_π(G|θ)| = |I_π(H|θ)|.This theorem translated the counting problem of I_π-character triple (G, N, θ) intothat of /^-character triple (H, N, 0), where H/N is a 7r-group, and gave a new sufficient condition for an /^-character which restricts irreducibly to a subgroup.